Question

Suppose we take a random sample X1,…,X5 from a normal population with an unknown variance σ2 and unknown mean μ.

Construct a two-sided 95% confidence interval for σ2 if the observations are given by

−1.25,1.91,−0.09,2.71,2.70

Answer 1

S.No		X	(X-x̄)	(X-x̄)2
1		-1.25	-2.446	5.98292
2		1.91	0.714	0.50980
3		-0.09	-1.286	1.65380
4		2.71	1.514	2.29220
5		2.7	1.504	2.26202
Σx		5.98	Σ(X-x̄)2=	12.7007
x̄=Σx/n		1.20	s2=Σ(x-x̄)2/(n-1)=	3.17518

	here n =	5
	s2=	3.175
Critical value of chi square distribution for n-1=4 df and 95 % CI				0.95
Lower critical value χ2L=		0.484
Upper critical valueχ2U=		11.143

for Confidence interval of Variance:
Lower bound =(n-1)s2/χ2U=(5-1)*(3.175/11.143)=		1.13979
Upper bound =(n-1)s2/χ2L=(5-1)*(3.175/0.484)=		26.24116
from above 95% confidence interval for population variance =(1.14<σ2<26.241)